Complete Fermi–Dirac integral
# Complete Fermi–Dirac integral

In mathematics, the complete Fermi–Dirac integral, named after Enrico Fermi and Paul Dirac, for an index j is defined by

    $F_{j}(x)={\frac {1}{\Gamma (j+1)}}\int _{0}^{\infty }{\frac {t^{j}}{e^{t-x}+1}}\,dt,\qquad (j>-1)$

This equals

    $-\operatorname {Li} _{j+1}(-e^{x}),$

where $\operatorname {Li} _{s}(z)$ is the polylogarithm.

Its derivative is

    ${\frac {dF_{j}(x)}{dx}}=F_{j-1}(x),$

and this derivative relationship is used to define the Fermi-Dirac integral for nonpositive indices j. Differing notation for $F_{j}$ appears in the literature, for instance some authors omit the factor $1/\Gamma (j+1)$. The definition used here matches that in the NIST DLMF.

## Special values

The closed form of the function exists for j = 0:

    $F_{0}(x)=\ln(1+\exp(x)).$

For x = 0, the result reduces to

$F_{j}(0)=\eta (j+1),$

where $\eta$ is the Dirichlet eta function.

